Optimal. Leaf size=192 \[ -\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}-\frac{3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{160 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
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Rubi [A] time = 0.25289, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2835, 2611, 3768, 3770, 2606, 266, 43} \[ -\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}-\frac{3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{160 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2611
Rule 3768
Rule 3770
Rule 2606
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^7(c+d x) \tan ^4(c+d x) \, dx}{a}-\frac{\int \sec ^6(c+d x) \tan ^5(c+d x) \, dx}{a}\\ &=\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac{3 \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx}{10 a}-\frac{\operatorname{Subst}\left (\int x^5 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac{3 \int \sec ^7(c+d x) \, dx}{80 a}-\frac{\operatorname{Subst}\left (\int (-1+x)^2 x^2 \, dx,x,\sec ^2(c+d x)\right )}{2 a d}\\ &=\frac{\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac{\int \sec ^5(c+d x) \, dx}{32 a}-\frac{\operatorname{Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sec ^2(c+d x)\right )}{2 a d}\\ &=-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac{3 \int \sec ^3(c+d x) \, dx}{128 a}\\ &=-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{3 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac{3 \int \sec (c+d x) \, dx}{256 a}\\ &=\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{3 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}\\ \end{align*}
Mathematica [A] time = 5.72551, size = 116, normalized size = 0.6 \[ \frac{-\frac{90}{\sin (c+d x)+1}-\frac{45}{(\sin (c+d x)-1)^2}-\frac{45}{(\sin (c+d x)+1)^2}+\frac{40}{(\sin (c+d x)-1)^3}+\frac{20}{(\sin (c+d x)+1)^3}+\frac{30}{(\sin (c+d x)-1)^4}+\frac{90}{(\sin (c+d x)+1)^4}-\frac{48}{(\sin (c+d x)+1)^5}+90 \tanh ^{-1}(\sin (c+d x))}{7680 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 180, normalized size = 0.9 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}+{\frac{1}{192\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{3}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{512\,da}}-{\frac{1}{160\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{3}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{384\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{3}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0931, size = 289, normalized size = 1.51 \begin{align*} -\frac{\frac{2 \,{\left (45 \, \sin \left (d x + c\right )^{8} + 45 \, \sin \left (d x + c\right )^{7} - 165 \, \sin \left (d x + c\right )^{6} - 165 \, \sin \left (d x + c\right )^{5} + 219 \, \sin \left (d x + c\right )^{4} - 421 \, \sin \left (d x + c\right )^{3} - 211 \, \sin \left (d x + c\right )^{2} + 109 \, \sin \left (d x + c\right ) + 64\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac{45 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{45 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07818, size = 517, normalized size = 2.69 \begin{align*} -\frac{90 \, \cos \left (d x + c\right )^{8} - 30 \, \cos \left (d x + c\right )^{6} - 12 \, \cos \left (d x + c\right )^{4} + 176 \, \cos \left (d x + c\right )^{2} - 45 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 45 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (45 \, \cos \left (d x + c\right )^{6} + 30 \, \cos \left (d x + c\right )^{4} - 616 \, \cos \left (d x + c\right )^{2} + 432\right )} \sin \left (d x + c\right ) - 96}{7680 \,{\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4095, size = 211, normalized size = 1.1 \begin{align*} \frac{\frac{180 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{5 \,{\left (75 \, \sin \left (d x + c\right )^{4} - 300 \, \sin \left (d x + c\right )^{3} + 414 \, \sin \left (d x + c\right )^{2} - 196 \, \sin \left (d x + c\right ) + 31\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac{411 \, \sin \left (d x + c\right )^{5} + 2415 \, \sin \left (d x + c\right )^{4} + 5730 \, \sin \left (d x + c\right )^{3} + 6730 \, \sin \left (d x + c\right )^{2} + 3515 \, \sin \left (d x + c\right ) + 703}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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